 By Maurice Heins

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Sample text

Show that if up) is the kth possesses a finite coefficient of the power series with sumf, on A(0; r), then (aP))nE~ limit ak, k E N, and f ( z ) = akz', IzI < r. What is the situation when A(0; r) is replaced by C? 12. Suppose that (ah)and (bk) are such that {k: ak# 0) and { k : br # 0) are bounded. Note that {k: ck # 0) is bounded. Suppose now that X,"akzk and Z," bkzk are power series subject only to the restriction that each converges on A(0; r), r positive. Let ck be defined as in the previous exercise.

Then the convergence set of C; akzk contains C(O; 1) - { I } . In fact, X? akzk converges uniformly on { [ z l I 1, (z - I1 > r } , 5. Power Series 0

They furnish methods of investigation of great value in the theory of differential equations, as we shall see later. It should be kept in mind that i n the study of power series the special case a, = 1 (geometric series) is very important in that it is an instrument of comparison about which we have very complete information. Let us turn to the question of the nature of the convergence set of a power series. It is easy to construct power series with convergence set (0) as well as power series with convergence set C.